Legendre molecular theories

Realistic intermolecular interaction potentials for mesogenic molecules can be very complex and are generally unknown. At the same time molecular theories are often based on simple model potentials. This is justified when the theory is used to describe some general properties of liquid crystal phases that are not sensitive to the details on the interaction. Model potentials are constructed in order to represent only the qualitative mathematical form of the actual interaction energy in the simplest possible way. It is interesting to note that most of the popular model potentials correspond to the first terms in various expansion series. For example, the well known Maier-Saupe potential JP2 (Sfli )) is just the first nonpolar term in the Legendre polynomial expansion of an arbitrary interaction potential between two uniaxial molecules, averaged over the intermolecular vector r,-, ... 

Here u is a unit vector oriented along the rotational symmetry axis, while in a spherical molecule it is an arbitrary vector rigidly connected to the molecular frame. The scalar product u(t) (0) is cos 0(t) in classical theory, where 6(t) is the angle of u reorientation with respect to its initial position. It can be easily seen that both orientational correlation functions are the average values of the corresponding Legendre polynomials ... 

Photoisomerization was studied from a purely photochemical point of view in which photo-orientation effects can be disregarded. While this feature can be true in low viscosity solutions where photo-induced molecular orientation can be overcome by molecular rotational diffusion, in polymeric environments, especially in thin solid film configurations, spontaneous molecular mobility can be strongly hindered and photo-orientation effects arc appreciable. The theory that coupled photoisomerization and photo-orientation processes was also recently developed, based on the formalism of Legendre Polynomials, and more recent further theoretical developments have helped quantify coupled photoisomerization and photo-orientation processes in films of polymer. 

This book is an introduction to quantum mechanics and mathematics that leads to the solution of the Schrodinger equation. It can be read and understood by undergraduates without sacrificing the mathematical details necessary for a complete solution giving the shapes of molecular orbitals seen in every chemistry text. Readers are introduced to many mathematical topics new to the undergraduate curriculum, such as basic representation theory, Schur s lemma, and the Legendre polynomials. 

The third equivalent formulation of classical mechanics to be briefly discussed here is the Hamiltonian formalism. Its main practical importance especially for molecular simulations lies in the solution of practical problems for processes that can be adequately described by classical mechanics despite their intrinsically quantum mechanical character (such as protein folding processes). However, more important for our purposes here is that it can serve as a useful starting point for the transition to quantum theory. The basic idea of the Hamiltonian formalism is to eliminate the / generalized velocities in favor of the canonical momenta defined by Eq. (2.54). This is achieved by a Legendre transformation of the Lagrangian with respect to the velocities. 

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